Solving Practical Problems Using Difference Equations or Numerical Differentiation

Using difference equations or numerical differentiation to solve more complex practical problems. This experiment will cover the following content:

Experiment 3: Interpolation and Numerical Integration

- Formulation of interpolation problems and solution methodologies

- Principles, advantages, and disadvantages of Lagrange interpolation (polynomial-based approach with basis functions)

- Fundamentals, strengths, and limitations of piecewise linear and cubic spline interpolation (smoothness and continuity considerations)

- MATLAB implementation of piecewise linear and cubic spline interpolation using built-in functions like interp1 for linear and spline for cubic interpolation

- Theoretical basis and MATLAB coding for trapezoidal and Simpson's integration formulas (vectorized implementations for efficiency)

- Error analysis in numerical integration formulas: understanding convergence order concepts

- Gaussian quadrature formulas (optimal node selection for polynomial exactness)

- Generalized integrals and multiple integrals (handling improper integrals and multidimensional cases)

- Applying interpolation and numerical integration techniques to solve more complex real-world problems

Experiment 4: Numerical Solutions of Ordinary Differential Equations

- Principles of Euler method and conceptual framework of Runge-Kutta methods (single-step vs multi-step approaches)

- Concepts of local truncation error and precision analysis

- MATLAB implementation of Runge-Kutta methods using ode45 solver, including solutions for differential equation systems and higher-order differential equations through variable substitution techniques